For which real value of a do the roots of the | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) For the roots of a quadratic equation $Ax^2 + Bx + C = 0$ to have opposite signs,the product of the roots must be negative,i.e.,$\frac{C}{A} < 0$.

Vedclass · Accepted Answer

$0 < a < 4$

Vedclass · Answer

(B) For the roots of a quadratic equation $Ax^2 + Bx + C = 0$ to have opposite signs,the product of the roots must be negative,i.e.,$\frac{C}{A} Given the equation $2x^2 - (a^3 + 8a - 1) x + (a^2 - 4a) = 0$,we have $A = 2$ and $C = a^2 - 4a$.The condition for opposite signs is $\frac{a^2 - 4a}{2} This implies $a^2 - 4a Factoring the expression,we get $a(a - 4) This inequality holds when $0 Since the discriminant $D = B^2 - 4AC$ is always non-negative for these values of $a$ (as $AC  0$),the roots are guaranteed to be real and distinct.Thus,the correct range is $0 < a < 4$.

For which real value of $a$ do the roots of the quadratic equation $2x^2 - (a^3 + 8a - 1) x + a^2 - 4a = 0$ have opposite signs?

Similar Questions

If $x = 2 + \sqrt{3}$,then what is the value of $x^3 - 7x^2 + 13x - 12$?

If the roots of the equations $ax^2 + 2bx + c = 0$ and $bx^2 - 2\sqrt{ac}x + b = 0$ are real,then:

If $3$ is a root of the equation $x^2 + kx - 24 = 0$,then $3$ is also a root of which of the following equations?

$A$ number is the reciprocal of the other. If the arithmetic mean of the two numbers is $\frac{13}{12}$,then the numbers are:

If ${x_1}, {x_2}, {x_3}$ are distinct roots of the equation $ax^2 + bx + c = 0$,then:

Vedclass Test Series

Exam Paper Generator

Online Exam Module